Blender won't copy pose in dope sheet and mirror paste it
About your copy-paste problem: maybe you've enabled the Copy Attribute addon, which seems to disable the normal copy-paste function in Pose mode and Dopesheet. If you have it enabled, use the copy-paste buttons on the horizontal header.

About your armature: First I personally prefer not to activate the Automatic Keyframe Insertion and the Active Keying buttons because I find it a bit disturbing as it creates keyframes as soon as you move a bone, but it's my own preference and maybe you feel comfortable with them.

Also, when you create key poses (in your case frames 0, 10, 20...) I think it's better to create keyframes for all the bones so you're sure that the flipped pose will be symmetrical and won't depend on what happens in the previous and next keyframes.

Now I think the main problem in your settings is your bone called Root.

First I don't understand the use you make of this bone. The root bone should be the hierarchy top parent, meaning parent of the hip, of the IK controllers, IK pole targets, etc... which is not the case here: You only parented Hips to Root, but what does Roots brings to the Hips function? It's not clear to me... So maybe first fix that.

Once done, most of the time, you won't need to use the root bone, except sometimes when you need to move the whole character.

So, what's exactly the bug that is happening when you copy-flippaste your bones? Actually it flippastes the left bones poses to their symmetrized right copies, and vice versa. But it also symmetrizes the pose of any other bones on their own X axis. That's what happens to Root: As you've created a pose for Root, it symmetrizes it on its X axis, and as its X axis points backward, it's flipped upward, and therefore it moves up Roots and its child Hips and all the Hips children!

So once you'll have changed the Root bone settings, and as you won't have any reason to move it, the problem won't happen any more.